Spiral gearing and gear teeth



Feb. 16,- 1943. N, TRBOJEVICH 2,311,006

SPIRAL GEARING AND GEAR TEETH Filed July 31, 1941 3 Sheets-Sheet 1INVENTOR .Feb.16,1943. 7 N. mama/1c 2,311,006

SPIRAL GEARING AND GEAR TEETH Filed July s1, 1941 3 Sheets-Shoot 2'5-2/5 INVENTOR Feb. I6, 1943. N. TRBOJEVICH' 2,311,006

sum GEARING AND emu TEETH Filed July :51, 1941 s Sheets-Sheet :5

INVENTOR HHU lvlcurlmilolvio DB3 [CH Nil)!" Patented Feb. 16, 1943UNITED STATES PATENT OFFICE 8 Claims.

The invention relates to a system of gearing comprising a smaller memberin the form of a helical pinion and a larger mating member in the formof a wheel having spiral teeth formed upon its face adjacent to the saidpinion.

Gearing of this general form is not new in the art, but I discovered anovel type of a helical pinion whereby the interferences between meshingteeth are avoided and the useful contact is correspondingly prolonged.

In particular, the new pinion and the mating wheel are characterized bythe fact that the transverse axes of their teeth form an acute anglerelative to their respective axes of rotation thus Figures 4 and 5 arediagrams explanatory of the principle of the tilted tooth cross axes.

Figures 6, 7, 8, 9, 13, 14, and 16 are geometrical diagrams used indeduction of the Equations 1 to 22 inclusive.

Figures 10 and 11 are two diagrams explanatory of the principle of polarsubnormals in the design of tooth spirals.

Figure 12 is a diagrammatic and perspective view of the finished gearteeth.

Figures 17 and 18 are diagrams explanatory of the design and the methodof hobbing the new pinion teeth.

Figure 19 is a diagram explanatory of the resulting in a non-symmetricalarrangement of i5 process of shaping the new gear teeth. the operatingtooth flanks in the planes of rota- As shown in Figures 1, 2 and 3, thedriven memtion. In the pinion this feature exemplifies itself ber is aringshaped wheel ll having an axis of in the novel arrangement of teethin which the rotation I2, a central bore I3 and a plurality of saidtransverse axes are offset from the axis of spiral teeth [4 formed inits plane face. Each rotation. 20 of the said teeth is longitudinallycurved along The object is to produce a helical pinion tooth the spirall5 thus being provided with an outand a mating spiral tooth, the latterhaving conwardly facing and convex bearing surface l6 cave and convexsides, in which the radii of curand an inwardly facing and concavesurface i1. vature are unequal at each side of each tooth, The spiralsl5 are of an ever increasing radius the arrangement being such that asmaller radius of curvature and are generated from a circle JA in onecooperates with a greater radius in the in the form of abridgedinvolutes of the circle. other and vice versa. It is by this means thatThe pitch surface is a hyperboloid of revolution the fouling ormutilation in the mating teeth I8. The transverse axes 20 of the saidspiral is avoided. teeth are inclined outwardly relative to the axis Theinvention is of a mathematical character of rotation l2 andsubstantially in a direction and is based upon the calculation ofcertain perpendicular to the said hyperboloid l8. It will hyperboloids,cones, spirals, and their curvatures. be noted that this tipping of theteeth, which is Much of this theoretical research rests upon mynecessary from a theoretical standpoint, does not previous discoveries,as particularly disclosed in impair the cross sectional area and thestrength my Patent No. 1,647,157 dated November 1, 1927. of the saidteeth. However, the details pertaining to the asymmetry The pinion 2| isrotatable about its axis 23 of the mating teeth are new. by means of anintegrally formed shank 22, the I further discovered new methods andtools for said axis being positioned in a plane tangent to themanufacture of the gear and pinion. The the circle JA and also in aplane perpendicular pinion is generated by means of a novel taper to theaxis ll of the wheel. This particular posihob. tioning refers only tothe preferred modification The practical object is to construct a gearof this invention and it is to be noted that my drive which may be usedin lieu of the present method of tooth construction is applicable to anyspiral bevel and hypoid gearing, e. g. in automoother positioning of theaxes as well. The said tive axles, and is simpler and cheaper tomanupinion consists of a cylindrical body upon which facture and adjust.a plurality of equi-spaced helical teeth 24 are In the drawings: formedall having a constant cross contour Figure 1 is the plan view of the newpinion throughout their lengths. The transverse tooth and gear showingthe gear teeth in cross section, axes 25 are inclined at an anglerelative to the l-l plane of Figure 2. axis of rotation similarly tothose of the mating Figure 2 is a sectional side view of Figure 1 wheel,thus resulting in the lopsided appearance taken through the broken plane2-2. of the said teeth in an end view as shown in Figure 3 is the endview of the pinion as viewed Figure 3. The pitch surface is a cone 28having from the point A, Figure 1. an apex at the point A.

The principle of the asymmetric formation of teeth in the gear andpinion is diagrammatically shown in Figures 4 and 5. In Figure 4 theasymmetric tooth it of the wheel I I contacts the corresponding piniontooth 24 with its concave side H at the point of contact 30. A normal 29is drawn at the said point to both cooperating tooth flanks I! and 28,and its points of intersection 3I with the pinion axis 23 and the point32 with the wheel axis l2 respectively, are noted. If the influence ofthe helix and spiral angles is disregarded for the time being, thedistance 30 3| of the normal corresponds to the radius of curvature ofpinion tooth flank 26 and the distance 3ll32 corresponds to the radiusof the gear tooth flank ll. The point to note is that as the slope ofthe flank l I relative to its axis of rotation I2 is increased, thenormal 29 swings upwardly into a new position denoted with the numeral33 and the radius of curvature of the pinion flank decreases while thatof the gear flank simultaneously increases, thus increasing the totaldivergence between the two mating surfaces rapidly; but if the saidslope is diminished, corresponding to a position 34 of the normal 29,the two radii tend to become more equal. For the purpose of thisinvention a considerable divergence in the curvature of the mating teethis required to avoid a mutilation of the said teeth and I discovered inthis method of tooth formation a simple and effective means for creatingsuch a divergence.

In Figure 5 the conditions prevailing at the convex sides iii of thegear teeth M are diagrammatically shown. In this case the radius ofcurvature 36 of the gear flank I6 is usually less than the radius 35 ofthe mating pinion flank 21 and both of the said flanks are convex. Thedivergence between the mating tooth faces is increased by diminishingthe slope the gear tooth face l6 relative to the corresponding axis ofrotation l2. It is thus seen that the theory requires that the slope onthe concave side of the gear tooth be increased and that on the convexside be decreased relative to the axis of rotation in order to create adivergence with respect to the pinion teeth. These two conditions arecompatible with each other and both are satisfied by inclining the toothaxes of both the gear and pinion teeth.

The matter of executing this principle in actual practice resolvesitself into a calculation of radii of curvature of the mating teethwhich is a matter of considerable mathematical difliculty when the factis considered that the pitch surfaces and the geometrical forms of theteeth themselves are not as yet known. Due to the lack of space only thebriefest mathematical treatment of this subject can and will be givenhere.

In Figures 6 and 7 the mating pitch surfaces of the pinion and gearrespectively are determined. An XYZ coordinate system is first selected,its Z axis coinciding with the gear axis l2, and the pinion axis 23being placed in the XY plane parallel to the X axis. Tentatively a pitchcone 28 (of the pinion) of a cone angle 6 and having an apex A in the Yaxis is selected. This cone is bodily rotated about the axis Z through avariable angle w. The geometrical envelope of this family of cones willbe the pitch surface of the mating gear.

The equation of the family of cones 28 is of the general form:

a function in four variables.

Its first partial differential with respect to w is next found From theEquations 1 and 2, after eliminating w from both, I obtain the equationof the pitch hyperboloid:

zt +y :z 0 cos 6 2 sin 6 (3) I now again partially differentiate theEquation 2 with respect to w:

After eliminating w from the Equation 4 by the use of the previousequations, I obtain two equations in :ryz.

2: 2 e sin 6 cos 6 e sin 6 (5) y =e cos 6 4. the line of contact betweenthe cone 28 and the hyperboloid I8 is a plane curve 39 lying in a planeparallel to the Z axis and specifically it is a hyperbola having thesame pair of asymptotes as the meridian hyperbola 38, see the Equation5. I shall now determine the tangent plane to both pitch surfaces at apre-selected point Q0 of the contact and analyze the kinematicalrelations in that plane.

Let R0 and To denote the pitch radii of the gear and pinion respectivelycontacting at the point Q0. After drawing a cone 40 tangent to thehyperboloid all along the pitch circle of a radius R0, I find its apexto lie in the Z axis at the point J and its cone angle to be '7. Then,the tangent plane is the triangle QoAJ. The skew arrangement of the twocones 28 and 4!! respectively is shown in perspective in Figure 8. InFigure 9 the tangent plane QtAJ is developed in the plane of paper. Inthat diagram the corresponding pitch cone radii of the gear and pinionare denoted with R. and r and they include an angle 1; with each other.The relation exists tan 7 tan 6=cos 1 for a angle of cooperating axes.

Let now the pitch cone radii R and r rotate about their respective axesJ and A in the same direction and at a definite ratio, and let n be thenormal at the point tangency Q0 of the two mating longitudinal toothspirals l5 and ll. The said normal intersects the line JA at a point Ewhich is the center of instantaneous rotation. This latter point may beenvisaged as a pitch point of an imaginary internal gear having an axisat J meshing with a pinion having an axis at A in the plane of paper.Thus the spirals I5 and ll are odontically conjugate to each other verymuch like two mating tooth curves in a pair of ordinary spur gears.

i t. lVlI'iUl ill .1.

AND

llZill-iANlSMS It can be proved mathematically that if the spiral 4| bean Archimedean spiral, the curve l5 will be an abridged involute havingan abridgment numerically equal to the polar subnormal AE of the firstspiral. I proved that point in certain publications of mine some yearsago. In Figures and 11, I shall show that the novel pinion possesses notone but two systems of Archimedean spirals in its tooth surfaces, onehaving a longer and the other a shorter polar subnormal. From above dataI determine the equations of the two systems (one for each side of theteeth) of the abridged lnvolutes in the gear teeth and calculate theirradii of curvature.

In Figure 10 the axial section of the new pinion is intersected by meansof the pitch cone generator 28 thus furnishing two sets of equispacedpoints of intersection with the teeth of the said pinion. The outwardlyfacing tooth flanks 26 are intersected at the points A'A"A"' of ashorter pitch A'A" etc., and the inwardly facing tooth flanks areintersected at the points B'B"B" of a longer pitch B'B" etc. Thus thereare two sets of constant pitch spirals lying in the surface of the pitchcone 28 when this process of intersection is completed all around thecircumference of the pinion. In plane development of the cone 28 thesecurves will appear as Archimedean spirals having a polar subnormal phaving a value in the first series:

r/ 21r sin6 where m is the number of teeth in the pinion.

A similar expression is found for the value of the polar subnormal ofthe other series by merely substituting the pitch B'B" for the pitch AAin the Equation 8.

In Fig. 11 (which is a projection of Figure 10) the pinion 2| isintersected by a plane tangent to the pitch cone 28 and this plane islaid in the plane of paper. The outside circumference of the pinion asintersected by the said plane gives the ellipse 42 and the points ofconvergence of the normals drawn to each of the two series ofArchimedean spirals are denoted with E and E" respectively thusdetermining the two polar subnormals p and 11" respectively. Theintersections of the said tangent plane with the teeth of the pinion arethe crescent shaped areas 43 and 43' forming a series of segments of anever increasing width and thickness and most important of all, ofshorter radii of curvature on their sides facing outwardly (the series AA etc.) and of longer radii on their sides facing inwardly. As a resultof this, the gear tooth spirals form two series of abridged lnvolutes l6and II respectively, both series having the same modifled base radius JAbut having different amounts of abridgment p" and p respectively.

In Figure 12 the completed teeth l4 of the novel gear I l are showndiagrammatically and in perspective. The pitch line 44 is the line ofintersection of the tooth l4 with the pitch hyperboloid l8, Figures 6and '7, and the area 45 adjacent to the said pitch line ismathematically identical with the tooth surface of the hyperboloidalgear shown in my Patent No. 1,647,157, Figures 2 and 3. Inasmuch as thepitch line 44 diagonally crosses the teeth [4, it follows that the saidteeth will have long dedenda 46 at their ends nearest to the center andlong addenda 41 at the outer circumference of the gear. This 76peculiarity limits the practical use of this type of gearing both as tothe available width of face of ,f' of the teeth as well as to thepracticable ratios of transmission. The latter should be greater thanabout four to one, according to my calculations.

In Figure 13 the relation between the circumferential velocities V1 andV: of the pinion and gear respectively is shown. From this relation theconjugate pitch radii are readily calculable as well as the slidingvelocity S. First, the sliding S:

Let now wi and w: be the angular velocities of the pinion and gear,

where m and 112 are the corresponding numbers of teeth in pinion andgear and Q is the ratio of transmission. Let further V0 denote thenormal velocity (in the direction of the common normal QoE) of bothpinion and gear. Then.

Vi cos a'=V2 cos a"=V0 (13) where a and a" are the helix or spiralangles of the pinion and gear. By a substitution from the Equations 10,11 and 12 I obtain:

from which n is determined. The relations also exist Qro cos a'=Ro cosa" 1) being the polar subnormal of the Archimedean spirals and r thepitch cone radius of the pinion.

In Figure 14 the method of deriving the equation and the radius ofcurvature R at the point of contact Q0 of the gear spiral I5 is shown.If an auxiliary coordinate system X'Y' is selected in addition to theoriginal XY system such that a o=QnA (18) where QoA is equal to thelength of the rack generator m, the equations of the spiral becomegreatly simplified and assume the form:

:cfe cos +a p sin rp} 1 --e 5111 rp-(lqn cos c in the X'Y' coordinatesystem. The mechanical generation of this spiral may be accomplished intwo different ways, viz: First, I roll the line m over the stationarycircle JE and connect the line 111. rigidly thereto; then, any point ofline m will describe an abridged involute l5 defined by the pair ofEquations 19. Secondly, I first construct the involute 48 from thecircle JE, draw a series of tangents thereto and scale off inwardly thedistance p upon each such tangent. The locus willagain be the same curveIS. The radius of curvature of the spiral I 5 at the point Qo is equalto R or the distance QoDo. Its exact value is calculable from theEquations 19 by a standard procedure known from differential calculus. Ihave found that value to be:

where n is the length of the normal QoE and a and ,0 have the values asindicated in the diagram.

Having thus found the curvature of the gear tooth spirals I proceed tomy next and last object in this analysis which is, to find the curvatureof the pinion tooth in any particular section thereof. From this I shalldetermine first the pressure angles of the pinion teeth at theiropposite sides and second, the angle to which the said teeth must betipped relative to the axis of rotation. The basic idea is that at thehollow sides of gear spirals the radius of curvature of the pinion toothin any plane section whatever must be less than the radius of the geartooth, as already stated.

In Figure 15 the method of calculating the radius of curvature r of theoutwardly facing pinion tooth flank 26 is shown. The calculation israther lengthy and consists of the following steps:

I first assume a base cylinder 49 coaxial with the pinion and scribeupon it a helix 50 of the same lead as the pinion. Next I draw a seriesof tangents i to the said helix thus obtaining the tooth surface 26 ofthe pinion which surface is an involute helicoid. I now intersect thesaid surface at its point Q0 by means of a plane parallel to the axis 23thus obtaining the equations of the curve of intersection 52 in the XYcoordinate system. These equations I differentiate twice to obtain thefirst and second derivatives (usually in a parametric form) from whichthe radius of curvature is obtained by using the same formula as before,in the case of the gear tooth spirals. After performing all thesecalculations I hav derived the formula: I

It cos t This ends the analysis of the conjugate dimensions in thepinion and gear including their radii of curvature. Once knowing thenormal pressure angle [3 on the outwardly facing flank 26 of the piniontooth (which side I call the coasting" side), I assume for the oppositeflank 21 (see Figure 16) a normal pressure angle as small aspracticable, say equal to zero, as the "driving" side of the pinion. Thetooth cross axis 25 will then bisect these two angles.

In Figures 17 and 18 the methods of designing and generating the newpinion are shown. The cross axes 25 of the pinion teeth 24 are alloffset from the axis of rotation 23 at the same distance thus makingthem all tangent to the circle 53, and equispaced. The pitch circle 54which also serves as the base circle for the driving flank 21) is nextdrawn, thus determining the pitch point 55. The first line of action tanB Q. 12'. I). (22) 56 (for the driving side) is drawn tangent to thepitch circle 55 and the second line of action 51 (for the coasting side26) is drawn symmetrically of the first line. This determines the basecircle 49 of the coasting side. The tooth flank 26 is an involute fromtop to bottom drawn from the base circle 49 while the opposite flank 21is partly an involute, at its addendum, and partly a radial line 58 atits dedendum.

The taper hob 59 has a pitch line 56 inclined relative to its axis ofrotation and a plurality of equispaced rack teeth 60 having pressureangles symmetrically disposed relative to the axis of rotation of thehob. The contours of the said cutting teeth 60 are straight lines inorder to generate involute curves in the pinion in all portions thereofwith the exception of the portion 6| which is a curve so generated thatit will be conjugate with the radial flank 58 of the pinion tooth andgenerate the same. The hob teeth are relieved in the same manner as anyordinary hobs are i. e. in the direction perpendicular to the hob axisthus producing the relieved contours 62. The cutting efficiency of sucha hob is equal at both sides of the pinion teeth in spite of theasymmetrical shape of the latter and the contour of the hob includingthe spacings and thicknesses of teeth remains unchanged after repeatedsharpenings.

In Figure 18 the method of generating the new pinions by means of thenew taper hob 59 in an ordinary hobbing machine is shown. The hob isusually single threaded, of the same hand as the pinion 2| to be cut andits axis 63 is tilted in a plane parallel to the axis of the pinion inthe same manner as when cutting ordinary helical pinions. The machine isalso geared up in the conventional manner and the cutting of the pinionproceeds in all respects similarly to the conventional method with theexception of the fact that the point of tangency between the hob and thepinion to be cut lies now in an offset plane 64 instead of theconventional centrally located plane 23.

In Figure 19 the method of generating the new spiral gear H isdiagrammatically illustrated. A helical gear shaping machine is usedwhich is provided with the so-called face cutting attachment which meansthat the axis 23 of the cutter spindle and the axis I! of the workspindle are disposed at an angle relatively to each other, usually at aright angle. The axes 23 and I2 are further so disposed as to be atdistance e, equal to the offset of the drive, from each other asmeasured in a projection perpendicular to the plane of the gear. Theshaping cutter 65 is a replica of the pinion shown in Figure 17 and hashelical teeth of the same hand and lead as the pinion. The cutter isreciprocated in a helical path by means of the aflixed helical guidescrew 66 and a corresponding nut 61, the latter being rotatable by meansof the worm gear 66. In this manner all the teeth H of the gear aregenerated in a continuous process and on their both sides.

It was suggested prior to this invention that the gear members of thisgeneral type could be produced by means of a hobbing process. I havefound that impossible to do because of the very unsatisfactory cuttingaction of the hob when so employed. The sliding velocity S, see Figure13, in this type of gearing is relatively small in comparison with thenormal or rolling velocity V0 for which reason the hob would mostly rolltogether with the work but would out very unti ill) MECHANiSMS (theSearch itimii satisfactorily if at all. Therefore, I generate the gearmember by means of the above described shaping process.

What I claim as new is:

5. A crown gear having a plurality of spiral teeth of a constant depthand a variable cross contour throughout their lengths formed in itsplane of rotation at one side of the said gear, in

1. A mating pair of gears consisting of a rotat- 5 which the said teethare asymmetrically shaped able cylindrical pinion having a plurality ofheliand are longitudinally curved to conform with cal teeth of aconstant cross section throughout a series of modified involutes havinga greater their length, and a rotatable spiral gear having degree ofmodification at their one side and with a plurality of longitudinallycurved teeth of ever another series having a less degree of modificac 'es g radii of curvature and a variable cross tion at the othersidesection throughout their lengths, in which the A c own gear having aplurality of spiral teeth of both members have transverse axes teeth ofa constant depth and a variable cross which are inclined atpredetermined acute angles n our and curva re hrou h their l n hsrelative to the corresponding axes of rotation di p d about an axis in aP e in ich ch for the purpose of increasing their relative dif the steeth has a longitudinally disp vergence from each other. concave andconvex surface of contact and in 2, A ati pair of gears consisting of at twhich the said surfaces are equispaced in a line able cylindricalpinion having a plurality of heli- Offset from the said is and a curvre1 cal teeth of a constant cross section, and a rotatt vely to the saidline at different radii of curvaable wheel having a plurality of spiralteeth of 2 ture at their concave and convex sides in order a variablecross section throughout their lengths, to simultaneously Contact a rackelement of in which the pinion contacts the said wheel in sta t pi chhaving asy t al t a line offset from the axis of rotation and diago- 7.A P of mating ea s Co p s a larger nally crossing the teeth of the saidwheel, and Wheel member having a plurality of ur e i which th teeth i bth members a asymteeth of a variable cross contour and curvaturemetrically formed relative to their respective throughout their lengthsand a s er 0 p o planes of rotation, but are substantially perpenmemberhaving a p a y of helical teeth f a di mr t t said line of t t, constantcross contour and curvature throughout 3, A crown wh l having aplurality of equitheir lengths in which further the teeth in both spacedteeth disposed in a plane, in which the 3 members are asymmetrical attheir pp said teeth are longitudinally formed along spirals sides andspecifically are abridged involutes of having ever increasing radii ofcurvature, in two different degrees of modification in the wheel whichthe said teeth have concave and convex member and two lnvolute helicoidsdeveloped contact surfaces formed at their opposite sides from twodifferent base heliXes in the P and in which the slope of inclination ofthe A pair of mating gears comprising a spiral concave surfaces is by adefinite amount greater gear havin a plurality of teeth of a va e thanthe slope of the convex surfaces, relative Cross on t ou h ut t eirlengths and asymto the said plane. metrically formed at their oppositesides and a 4. A crown gear having a rotatable disk shaped Cylindrical pn having a p u a y of elical body and aplurality of equispacedlongitudinally 40 teeth of a constant cross section throu ou curvedteeth formed in the face of the aid di their lengths and alsoasymmetrically formed at in which the transverse axes of the said teeththe Opposite sides thereof, in which the tOOt are outwardly divergentrelatively to the axis cross contouls in both members are convex at ofrotation of the gear in such a manner that all points d n w ch e ss ours of each of the said axes is substantially perpent gear teeth areconju a e to a m a f e dicular to a hyperboloid of revolution coaxialwith the said gear and having a gorge radius on the same side of thegear upon which the said spiral teeth are formed.

from the pinion member when the said lamina is rotated in a timedrelation in plane transverse of the said gear teeth.

NIKOLA TRBOJEVICH.

